Chapter 7 · Proportional Reasoning

7.2–7.3 Ratios and Proportion

Five photos of the same tiger, stretched by different amounts. Three of them still look right. Two look wrong. The reason is a ratio — and once you can name it, you can predict which stretch will look wrong before you even see it.

🎯 By the end of this you can…

✓Write a ratio and name its terms
✓Reduce a ratio to its simplest form using the HCF
✓Decide whether two ratios are "proportional" to each other
✓Explain why equal DIFFERENCES don't make two ratios proportional, but equal FACTORS do

💡 The idea

A ratio compares two quantities by division, not subtraction

The ratio of width to height of a 60 mm × 40 mm image is written $60 : 40$ . The numbers 60 and 40 are the ratio's terms. Five images, stretched by different amounts from the same original:

📐 Same simplest-form ratio → looks similar

60×40 mm

3:2

40×20 mm

2:1

30×20 mm

3:2

90×60 mm

3:2

60×60 mm

1:1

A, C, D → 3:2 (proportional!)B → 2:1 E → 1:1

📋 Why A, C and D look right, and B and E don't

1
Reduce each image's width:height ratio to simplest form by dividing both terms by their HCF.
2
A = 60:40 → simplest 3:2. C = 30:20 → simplest 3:2. D = 90:60 → simplest 3:2. All three match.
3
B = 40:20 → simplest 2:1. E = 60:60 → simplest 1:1. Neither matches 3:2 — both look distorted.

✂️ Simplify 60 : 40

⭐ 5 XP

HCF of 60 and 40 is 20.

1 / 2

✂️ Simplify 40 : 20

⭐ 5 XP

HCF of 40 and 20 is 20.

1 / 2

⚠️ The classic trap

Image B's width and height are both exactly 20 mm less than image A's — same DIFFERENCE, 20 mm. It still looks wrong. A ratio compares by dividing, not subtracting: B's height was halved (40→20) while its width wasn't (60→40 is not a halving), so the SHAPE changed even though the gap stayed the same number.

💡 The idea

Two ratios are proportional when they reduce to the same simplest form

We write $a : b :: c : d$ (read "a is to b as c is to d") when $a : b$ and $c : d$ share the same simplest form. A quick test without reducing first: cross-multiply — $a : b :: c : d$ exactly when $a \times d = b \times c$ .

✏️ Worked example

Are 4:7 and 12:21 proportional?

1
Cross multiply: $4 \times 21 = 84$ and $7 \times 12 = 84$ .
2
Both sides equal — so yes, $4 : 7 :: 12 : 21$ .

✅ Answer: $4 : 7 :: 12 : 21$

🎮 True or false — the book's own statements

⭐ 0 XP🔥 streak 0

True or false:

4 : 7 :: 12 : 21

True or false:

8 : 3 :: 24 : 6

True or false:

7 : 12 :: 12 : 7

True or false:

21 : 6 :: 35 : 10

True or false:

12 : 18 :: 28 : 12

True or false:

24 : 8 :: 9 : 3

0 of 6 finished

🤔 Which of these is proportional to 4:9?

🎮 Our own practice — fill in proportional to 15:20

⭐ 0 XP🔥 streak 0

Find

6 : ?

proportional to

15 : 20

Find

9 : ?

proportional to

15 : 20

Find

30 : ?

proportional to

15 : 20

0 of 3 finished

🏅

You finished this lesson!

Ratios compare by dividing, and "proportional" means the same simplest form. Next: using that idea to actually SOLVE for a missing number — the method ancient Indian mathematicians called the Rule of Three.

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